With respect to the usual notations of an $A.P.$,if $a=5, d=3$ and $T_n=50$,find $n$ and $S_n$.

The production of a $TV$ factory increases equally every year. Its production in the $3^{rd}$ year was $600 \, TV$ and its production in the $7^{th}$ year was $700 \, TV$. Find the production in the $1^{st}$ year,in the $10^{th}$ year,and the total production in $7$ years.

With respect to the usual notations of an $A.P.$,if $T_{n} = 28$,$S_{n} = 144$,and $n = 9$,find the first term $a$.

For an $A.P.$,the $7^{th}$ term is $-1$ and the $16^{th}$ term is $17$. Find the general term of the $A.P.$

The $A.P.$ with the first term $-3$ and the common difference $4$ is $ \ldots \ldots \ldots \ldots . $

The sum of the first $n$ terms of an $AP$ whose first term is $8$ and the common difference is $20$ is equal to the sum of first $2n$ terms of another $AP$ whose first term is $-30$ and the common difference is $8$. Find $n$.